As soon as the tributes begin to the filmed, the Games truly begin. The arena is only the final stage of the Games (although, Katniss soon discovers that the Games are never over for the winners). Being on camera allows the tributes to gain support and sympathy, making them more appealing to sponsors. Also, since the Capital wants a big show, the more dynamic they are on camera, the more likely they are to get attention and sponsors. We discover that the key to surviving isn't necessarily being the best in the arena, but the most liked, and therefore the most sponsored. Since the Games are all about the show, being filmed also means that if you aren't doing something entertaining, the Game Makers are likely to send something entertaining to you.
Being filmed makes the entire Games an act, or illusion. The tributes are only pawns, however, and the information they have is limited. It seems like this would make it harder to accept their fate. First, they are under the illusion that their is a "winner," which Katniss discovers to be false. Also, they are often mislead about their popularity, and therefore odds of winning. The tributes from the wealthier districts tend to believe themselves to have higher odds, but as the Games progress, this also proves to be false. Finally, since everything is set up as a "show," especially in the pregame ceremonies, the reality of death, murder, etc. hits the Tributes hard once they are in the arena. Up until this point, they have been viewing the Games as just that - games. Suddenly, it all becomes real, and not all the Tributes are able to adjust accordingly.
In the Games everything is televised LIVE. So anyone can see you. they judge you by the way that you act and that will determine if you get more sponsers or not. And getting a lot of sponsors is a neccesity in the Games to help your survival.
Katniss did not want to appear weak like a little girl. She didn't want people to see her that way because that was not who she was. She acted the way that she did for her sister Prim and her mother. So that they may not lose hope but keep faith that Katniss may come home once again. That she will survive the Hunger Games and live with them in a wonderful home. She did it so that they could be safe
The camera is an essential part of the game. To an extent, the game becomes about illusions. In stark contrast to the very harsh life and death reality of the game where Katniss has very little control, she has much more control over how the viewers at home see her as a competitor. She can shape herself into a competitor that sponsors want to help regardless of her true emotions. It mirrors the contrast between life in the distrcits and life at the Capitol. Things as they are vs. things as they appear to be.
The camera, I think, made it easier for Katniss to accept her fate. She used the camera as a tool both to show her rage to the capital and to reach out to the districts. She knew that her skills alone would not get her out a live and that she had to depend on others- something that was extremely difficult for her. So the camera also forced her to change. It forced her into a new world where she could not depend only on herself. The camera was a constant reminder of the things she couldn't control and the things she could attempt to manipulate.
It makes it harder for them to accept their fates. Because they were constantly watched, they couldn't show weakness. Panem was watching, the sponsors were watching, and the gamemakers were watching. The way Katniss survived was by feigning emotion for Peeta so that the nation could love them just as much as they "loved" each other.